Average Error: 30.1 → 0.2
Time: 41.6s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r8311697 = x;
        double r8311698 = 1.0;
        double r8311699 = r8311697 + r8311698;
        double r8311700 = sqrt(r8311699);
        double r8311701 = sqrt(r8311697);
        double r8311702 = r8311700 - r8311701;
        return r8311702;
}
↓
double f(double x) {
        double r8311703 = 1.0;
        double r8311704 = x;
        double r8311705 = r8311704 + r8311703;
        double r8311706 = sqrt(r8311705);
        double r8311707 = sqrt(r8311704);
        double r8311708 = r8311706 + r8311707;
        double r8311709 = r8311703 / r8311708;
        return r8311709;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.1

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.9

    \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\]

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))